The growth of III-V semiconductors on silicon would allow the integration of their superior (opto-)electronic properties with silicon technology. But fundamental issues such as lattice and thermal expansion mismatch and the formation of antiphase domains have prevented the epitaxial integration of III-V with group IV semiconductors. Here we demonstrate the principle of epitaxial growth of III-V nanowires on a group IV substrate. We have grown InP nanowires on germanium substrates by a vapour-liquid-solid method. Although the crystal lattice mismatch is large (3.7%), the as-grown wires are monocrystalline and virtually free of dislocations. X-ray diffraction unambiguously demonstrates the heteroepitaxial growth of the nanowires. In addition, we show that a low-resistance electrical contact can be obtained between the wires and the substrate.